In C.D.Sogge's *Fourier Integrals in Classical Analysis* pp.128-129, he proved Lemma4.2.3(Tauberian Lemma):

**Lemma.** Let$g(\lambda)$ be a piece-wise continuous tempered function of $\mathbb{R}$. Assume that for $\lambda>0$
$$|g(\lambda+s)-g(\lambda)|\leq C(1+\lambda)^a,0<a\leq1$$
Then if $\hat{g}(t)=0$ when $|t|\leq1$(The hat means Fourier transform) we have:
$$|g(\lambda)|\leq C(1+\lambda)^a$$#

by using an identity:

$$|G(\lambda)|=|(G'*\psi)(\lambda)|\leq C(1+\lambda)^a\int|\psi(s)|(1+|s|)^ads\leq C(1+\lambda)^a$$

My confusion is that when the $\hat{\psi}(t):=\frac{\eta(t)}{it}$ where $\eta\in\mathcal{S}$ satisfying $\eta(t)=0$ when $|t|\leq\frac{1}{2}$ and $\eta(t)=1$ when $|t|>1$. It could be that $\psi$ is not integrable at all since $\frac{1}{t}$ is not necessarily integrable at infinity. *So how could this identity holds?*

What is more, in the same book, pp.127 in the proof to Theorem4.2.1 the same argument arise. Altough this can be argued using oscillatory integral in Chap1.

**Remarks from Professor** It could be found in Hormander's PDO Vol.3 this Lemma holds under a more restrictive assumption, and it suffices for later applications.

**My question is whether the identity above can be argued correctly?** If so, what technique should be used?(I will be deeply appreciated if a detail explanation is given.)

Thanks.

The Analysis of Linear Partial Differential Operators$\endgroup$